The purpose of this note is to give upper bounds (assuming P different from NP) on how far the generalizations of Skolem sequences can be taken while still hoping to resolve the existence question. We prove that the existence questions for both multi Skolem sequences and generalized Skolem sequences are strongly NP-complete. These results are significant strengthenings and simplifications of th...

Let G be a graph and f : V (G) → {1, 2, 3, . . . , p+ q} be an injection. For each edge e = uv and an integer m ≥ 2, the induced Smarandachely edge m-labeling f∗ S is defined by f ∗ S(e) = ⌈ f(u) + f(v) m ⌉ . Then f is called a Smarandachely super m-mean labeling if f(V (G))∪ {f∗(e) : e ∈ E(G)} = {1, 2, 3, . . . , p+ q}. Particularly, in the case of m = 2, we know that f ∗(e) =    f(u)+f(v) ...

The chordality of an undirected graph G, which is not acyclic, is defined as the length of the longest induced cycle in it. The chordality of an acyclic graph is defined to be 0. We use Cl (l ≥ 3) to denote a cycle of length l. An induced cycle is called a hole. A hole is an odd hole if its length is odd and is an even hole otherwise. Odd-chordality of a graph is the length of the longest odd h...

The odd-girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function f( ) for each : 0 < < 1 such that, if the odd-girth of a planar graph G is at least f( ), then G is (2 + )-colorable. Note that the function f( ) is independent of the graph G and → 0 if and only if f( )→∞. A key lemma...

A signed graph is a pair (G,Σ) where G is a graph and Σ is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of Σ. A blocking pair of (G,Σ) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts i...

Motivated by the famous Skolem-Mahler-Lech theorem we initiate in this paper the study of a natural class of determinantal varieties which we call Vandermonde varieties. They are closely related to the varieties consisting of all linear recurrence relations of a given order possessing a non-trivial solution vanishing at a given set of integers. In the regular case, i.e., when the dimension of a...

Motivated by the famous Skolem-Mahler-Lech theorem we initiate in this paper the study of a natural class of determinantal varieties which we call Vandermonde varieties. They are closely related to the varieties of linear recurrence relations of a given order possessing a non-trivial solution vanishing at a given set of integers. In the regular case, i.e. when the dimension of a Vandermonde var...

An odd dominating set of a simple, undirected graph G = (V,E) is a set of vertices D ⊆ V such that |N [v]∩D| ≡ 1 mod 2 for all vertices v ∈ V . It is known that every graph has an odd dominating set. In this paper we consider the concept of connected odd dominating sets. We prove that the problem of deciding if a graph has a connected odd dominating set is NP-complete. We also determine the exi...

Graph theory plays a significant role in variety of real-world systems. concepts such as labeling and coloring are used to depict processes relationships material, social, biological, physical, information Specifically, graph is communication network addressing, fault-tolerant system design, automatic channel allocation, etc. 2-odd assigns distinct integers the nodes <img src=image/13427595_01....

Let G be a graph and let c : V (G) → ({1,...,5} 2 ) be an assignment of 2-element subsets of the set {1, . . . , 5} to the vertices of G such that for every edge vw, the sets c(v) and c(w) are disjoint. We call such an assignment a (5, 2)-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph G is the length of the shortest odd ...